Theoretical Model for Normal Impact between Dry Sphere and Liquid Layer with Considerable Thickness

Detailed understanding on the impact mechanism with presence of liquid is helpful for elucidating complex behaviors of wet particle flows. This paper has extended elastohydrodynamic theory by coupling the physical force models for the wet normal impact with a liquid layer of considerable thickness. The contact is divided into penetration, contact, and rebound stages. Three critical Stokes numbers are developed for each stage to predict the sphere rebound behavior. The dependence of the critical Stokes numbers on the layer thickness, sphere radius and the liquid viscosity is also studied. When the layer thickness and liquid viscosity ranges within a low level, most of the kinetic energy is dissipated in the contact stage. With further increase of these two parameters, the energy dissipation in the penetration stage becomes predominant. Finally, we developed an equation to predict the restitution coefficient of the wet normal impact with acceptable accuracy.


INTRODUCTION
It is common in real-world applications that a small amount of liquid is added into particulate processes, for example, particle synthesis and surface modification, pharmacy, and food processing.The presence of liquid makes the particles coated with liquid layers or agglomerated by liquid bridges, which results in the particulate flow behaviors greatly different from that of dry system (Seville andClift, 1984, Mitarai andNori, 2006;Zhu et al., 2013;Zhou et al., 2013;Zhou et al., 2016).Particle-particle and particlewall impacts play key roles in such flows.A detailed understanding on the individual impact with liquid is helpful to reveal the mechanisms of these flows.
Theoretical mechanisms of impacts with liquid has been an increasing interest but still in controversy, because unlike impacts between dry bodies, the introduction of viscous damping from liquid contributes to the complexity of physical description of the impact.In the literature, there are three theoretical approaches to study wet impact generally, which are (1) Direct Numerical Simulation (DNS) model of particle interface (Jain et al., 2012); (2) physical modeling of different forces acting on the particle during impact (Antonyuk et al., 2009), and (3) elastohydrodynamic theory or lubrication theory (Davis et al., 1986).
With DNS approach, Jain et al. (2012) developed a combination of the Volume of Fluid (VOF) and Immersed Boundary (IB) model to simulate particle impact on thin liquid films and revealed detailed behaviors.Tobias and Jochen (2012) developed interface-resolving numerical simulations of particle-particle impacts for revealing the impact mechanisms.Since the DNS approach itself is a challenge, it is out of the scope of the present work.
By physical modeling of forces, Antonyuk et al. (2009) illustrated the energy balance during impact and presented a reasonable prediction of restitution coefficient.However, one weak point of their model is that it involves lots of assumption parameters, especially for the contact force model.
The elastohydrodynamic theory or lubrication theory, originally proposed by Davis and his colleagues seems an elegant method.Davis et al. (1986) studied theoretically a head-on impact of a sphere on a plate immersed in a viscous liquid and developed elastohydrodynamic theory to quantify rebound characteristics.According to this theory, the restitution coefficient, which is defined as the ratio of sphere velocity after impact to that before impact, can be described by Stokes number, St d and critical Stokes number St dc .St d represents the ratio of the inertia of sphere to the viscous forces exerted by the liquid layer.St dc is the critical value, below which, the kinetic energy of sphere is fully dissipated and no rebound would be observed (Barnocky and Davis, 1988).(2) where m is the sphere mass, v is the impact velocity, μ is the liquid viscosity, a is the sphere radius, x 0 indicates the region where the elastohydrodynamic theory is applicable.
It represents the distance between sphere tip and plate surface.Basically, x 0 /a should be much less than unity.x 1 is the elasticity length scale for deformation, representing the distance from the plate surface when the sphere restores enough elastic energy to rebound through deformation.θ = (1 -ν 1 2 )/πE 1 + (1 -ν 2 2 )/πE 2 , with E i and ν i representing Young's modulus and Poisson's ratio for the sphere (i = 1) and plate (i = 2) respectively.
The elastohydrodynamic theory was then applied in several experiments to characterize impact processes.Joseph et al. (2001) performed a normal impact of a particle attached to a pendulum with a plate immersed in viscous liquid and found that no rebound is observed for St d < St dc .When St d > St dc , the restitution coefficient, e wet , increases sharply before reaching a constant value.Similar experiments for immersed impact were also carried out by Gondret et al. (1999), Gondret et al. (2002) and Lundberg et al. (1992), with consistent results with the elastohydrodynamic theory.
Besides the immersed impacts, Davis et al. (2002) affirmed the applicability of the elastohydrodynamic theory to the impact with thin liquid layer, where the conditions are restricted to h << a, where h is the layer thickness.An equation was proposed to predict e wet with a simple form.e dry is the restitution coefficient for the dry impact.It is used to account for the energy dissipation in the solid.
Although the elastohydrodynamic theory has been successfully applied to various wet impacts (immersed impact and thin-layer impact), it is not appropriate for characterizing the impact between the sphere and liquid layer with considerable thickness.Briefly, the elastohydrodynamic theory neglects the effects of layer thickness in the energy dissipation, manifesting as the absence of layer thickness in the definition of St d .However, it has been proved that the layer thickness plays an important role in the impact (Antonyuk et al., 2009;Crüger et al., 2015;Ma et al., 2015).Therefore, in our early work (Ma et al., 2013), we proposed a modified Stokes number, St m to incorporate the effects of layer thickness: In the present paper, we extended elastohydrodynamic theory to thick-layer impact system by coupling the physical modeling of forces for further physical insight into the impact mechanism.The objective of the theory is two-fold: i) to predict whether the sphere would rebound after the impact; ii) to estimate the restitution coefficient.The experimental data used in the following was obtained through our previous work (Ma et al., 2015).

THEORETICAL DEVELOPMENT
Critical Stokes number, St mc , is usually used to predict whether the sphere would rebound subsequent to a normal impact.It is the St m at which e just drops to zero.In order to estimate St mc , we divided the whole impact process into three stages: 'penetration', 'contact' and 'rebound' as illustrated in Fig. 1.The centroid of the sphere whose tip just contacts the liquid surface is set as original point and that the downward direction is chosen as the plus direction.
The penetration stage starts from the liquid layer surface to 0.01a from the plate surface.The contact stage begins at 0.01a and ends at the elasticity length scale for deformation, x 1 .Within this region, the elastohydrodynamic theory is applicable (the prime condition for the theory is x 0 << a.That is why we set 0.01a as the initial point of contact stage).The rebound stage covers the process from the formation of liquid bridge to the maximum bridge length.
In terms of impact process, the sphere velocity could drop to zero at any one of the three stages.Therefore, three critical Stokes numbers have to be developed, aiming at full absorption of kinetic energy in different stages.Previously, several assumptions have to be made as follows: (1) The liquid layer is Newtonian fluid, namely, its rheological behavior is constant throughout the impact process.(2) A rough calculation on the magnitude of viscous force has been made for the case that the kinetic energy of sphere is fully dissipated.It is far larger than the sphere gravity.Thus, we neglected the effects of sphere gravity in the modeling process.
(3) Surrounding air has negligible effect on the impact dynamics.

Penetration:
During the penetration stage, the kinetic equations can be approximated as: where m is the sphere mass, v(t) is the relative velocity of the sphere toward the plate.F D and F vis are the drag force and viscous force imposed by the liquid respectively.
Here, ρ l is the liquid density, A D = π(2ax -x 2 ) is the characteristic area, C D is the drag coefficient which is evaluated using the following equation: Eq. ( 9) is applicable for a wide range of Re p (0 < Re p < (2-4) × 10 5 ) (Antonyuk et al., 2009).Re p = d pe vρ l /μ, where d pe is the equivalent diameter of the immersed part of the sphere in the liquid layer.In order to simplify the modeling process, we used the length of sphere immersed in liquid, x, as the characteristic length of Reynolds number, instead of d pe .Thus, the new Reynolds number is Re x = vρ l x/μ.Based on Re x , a simplified form of Eq. ( 9) is given as follows: 18 Re   9) and ( 10).studied, the predictions of C D from both equations agree well.It implies that Eq. ( 10) could provide acceptable C D in spite of its much more simplified form when compared with Eq. ( 9).
According to Adams and Edmondson (1987), the viscous force F vis can be found as: Therefore, dividing Eq. ( 7) by Eq. ( 6) to eliminate the time term and employing the boundary condition v = v 0 at x = 0 and v = v 1 at x = h -0.01a for integral, yields 2 0 2 1 0 1.5 ln 4.5 ( 4 ) where x 0 = 0.01a.Through rearranging Eq. ( 12), a correlation that involves St m can be obtained as Eq. ( 13).
If the kinetic energy of the sphere is fully absorbed in this stage, v 1 = 0.The critical Stokes number St mc1 is

Contact:
According to the elastohydrodynamic theory (Davis et al., 2002), the reduction of sphere velocity in the contact stage is given: v 2 is the sphere velocity at the beginning of the rebound.It can be calculated by combining Eqs. ( 13) and ( 15) as: ) If v 2 = 0 but v 1 ≠ 0, the kinetic energy of sphere is dissipated in the contact stage, where St d < St dc (Barnocky and Davis, 1988) is met.

Rebound:
As can be seen from Fig. 1, due to the effect of liquid inertia, the liquid would propagate away with higher amplitude than layer thickness during the penetration and contact stages.It results in that the amount of liquid acting on the sphere in the rebound process is much smaller.In this regard, the main resistance acting on the sphere could only be liquid bridge force instead of drag force and viscous force.
The liquid bridge force could be divided into dynamic force and static force.For a dynamic bridge, the former is much larger than the latter (Ennis et al., 1990;Pitois et al., 2000).Therefore, in the present paper, we incorporated the dynamic liquid bridge force into the model.
According to Matthewson (1988), for a thin-layer impact, the dynamic liquid bridge force can be approximated as: ( 1 ) H is the distance from the plate surface at wetted radius r, D is the distance between the sphere tip and plate surface (Fig. 3).
We estimated H-D as layer thickness h.The length that liquid bridge force imposes over is L max which is the maximum length of the liquid bridge throughout the impact.Based on the previous experimental work, we plotted L max Fig. 3. Geometry of liquid bridge.
as a function of St m in Fig. 4. St m is plotted in a logarithmic way to cover a wide range of values where the liquid viscosity ranges from 7 mPa.s to 2040 mPa.s and layer thickness in the range of 0.5 mm to 3 mm.As can be seen, taking St m = 210 as the cut-off point, L max increases first then falls algebraically with St m .The non-linear variation of L max corresponds to two bridge status: rupture and nonrupture.When St m < 210, the liquid bridge force is large enough to dissipate all the kinetic energy of sphere, thus no ruptures were observed.The decline of resistance associating with the increasing of St m lowers the decay rate of kinetic energy, therefore causing an increase of L max .With further increase of St m beyond 210, the liquid bridge would rupture before the sphere is brought to rest, when the kinetic energy of sphere E r exceeds the rupture energy of bridge E rup .The increasing of St m reduces E rup , causing the point where E r > E rup occur in advance, which results in a decline of L max .It is consistent with the demonstrations of Pitois et al. (2001) and Lian et al. (1993).Based on the experimental data, two equations are fitted for both cases respectively: max 0.0146 ln( ) 0.046 On the basis of kinetic equations, Eqs. ( 20) and ( 21) are obtained for the rebound stage: If the sphere velocity is brought to 0 in the rebound stage, v 3 = 0. Thus, the critical Stokes number in this stage, St mc2 can be achieved by combining Eqs. ( 13), ( 15) and ( 21 ) Basically, whether the sphere would rebound subsequent to a normal impact mainly depends on the comparisons of St m with three critical Stokes numbers.If St m > St mc1 , the sphere would pass the first stage with certain velocity.Then calculate the sphere velocity after penetration stage, v 1 using Eq. ( 12) and compare St d with St dc based on v 1 .If St d > St dc , the sphere could depart from the plate after contact stage.Finally, compare St m with St mc2 that is calculated with v 2 from Eq. ( 16).If St m > St mc2 , rebound occurs.Generally, the expectation that the sphere rebounds out of liquid layer must full fill the fact that the Stokes numbers are larger than the critical values for all the three stages.If not, the sphere velocity would be brought to zero in the corresponding stage.

Critical Stokes Number
Fig. 5 plots a comparison between experimental and theoretical St mc .As can be seen, the experimental data, in spite of their scatter, shows reasonable agreement with the theoretical predictions.The scatter may be attributable to the negligence of liquid inertia in the modeling process.In the present work, the effect of liquid inertia is only used to determine the resistance involved in the rebound stage.However, from rough observation, it can be confirmed that the liquid inertia also influences the penetration stage.Therefore, improving the physical models by incorporating the liquid inertia could enhance the predictability of the overall models.
Of significant practical interest is realizing the effects of various impact parameters on St mc which concerns whether a sphere would stick to the plate after impact.We used Eqs.( 14) and ( 22) to calculate St mc1 and St mc2 and study how they change with physical parameters.
Fig. 6 shows the critical Stokes number for the penetration stage, St mc1 , as a function of layer thickness, h and sphere radius, a.The variation of St m is also plotted in Fig. 6 for comparison.As can be seen, St mc1 decreases with the layer thickness while increases with the sphere radius.As h increases within a low level, St m is larger than St mc1 , implying that the sphere would go through the penetration stage with certain kinetic energy (Fig. 6(a)).With further growth of h, St m decreases more sharply than St mc1 .When h increases beyond a critical point, St m is smaller than St mc1 .The sphere velocity would be brought to zero in this stage and no rebound can be observed.The critical point inversely depends on the liquid viscosity.Opposite trend is found regarding the effects of sphere radius (Fig. 6(b)).The sphere inertia decreases with a, leading to a smaller St m as compared with St mc1 and full dissipation of kinetic energy in this stage.Until the radius increases beyond a critical point, St m exceeds Fig. 8 provides a comparison of the percentages of initial impact energy that are dissipated in the penetration, contact and rebound stages as a function of layer thickness and liquid viscosity.When the layer thickness ranges below 0.3 mm, the energy dissipated in the contact stage is the largest, indicating a predominant role of lubrication force (Davis et al., 1986).The percentage for the rebound stage is the smallest throughout.It implies that liquid bridge force makes less contribution to the energy dissipation as compared with drag force, viscous force and lubrication force.As the thickness further increases, the dissipative percentage in the penetration stage sharply increases, whereas the percentages of contact stage and rebound stage decrease monotonically.When the layer thickness exceeds 1.6 mm, the percentage of dissipative energy in the rebound stage becomes zero, implying that the sphere velocity is reduced to zero in the contact stage.At this point, most of kinetic energy is dissipated in the penetration stage.Similar trends can be observed considering the effects of liquid viscosity (Fig. 8(b)).Note that, the contribution of rebound stage and contact stage to the energy dissipation increases first with the liquid viscosity then decreases.Taking the contact stage as an example, the percentage of energy loss in this stage is defined as: R = (v 12 -v 2  energy loss in the contact stage still increases with liquid viscosity.This indicates that although v 1 decreases with increasing liquid viscosity, v 2 decreases more.Therefore, it can be concluded that the energy dissipation caused by the lubrication force (contact stage) is more sensitive to the liquid viscosity as compared with the penetration stage.
Based on the same reason, similar conclusion may be drawn regarding the rebound stage.

Restitution Coefficient
Regarding the impact on a liquid layer with large thickness, the effect of liquid bridge has to be considered since it also makes contribution to the energy dissipation.Therefore, the restitution coefficient in the present paper is defined as the ratio of the sphere velocity at the bridge rupture, V r , to the impact velocity, V i .
Due to the inapplicability of existing models in characterizing the "thick-layer impact" (Ma et al., 2015), we tried to derive a new equation to predict restitution coefficient based on St m .Fig. 9 plots the dependence of restitution coefficient, e on St m for normal impact with different velocities.As can be seen, the data shows less scatter when compared to that scaling with St d (Ma et al., 2015), implying a good applicability of St m in describing the "thick-layer impact".Moreover, from a qualitative perspective, Fig. 9 shares some similarities with those obtained previously for thin-layer impact (Davis et al., 2002;Kantak and Davis, 2004;Gollwitzer et al., 2012).Thus, the equation for "thick-layer impact" can be given in analogy with Eq. ( 4) by incorporating the modified Stokes number St m and St mc .It is worth pointing out that e dry in Eq. ( 4) serves as a reduction factor to characterize the energy dissipation in the solid.Basically, this factor should involve all the other factors that cause energy dissipation except viscous damping from liquid.For the impact with liquid, using e dry as the reduction factor is not appropriate, because the presence of liquid can not only cause energy loss through viscous force, but also affect the pressure profile over the sphere surface, thereby changing the deformation and rebound velocity of the sphere.In this regard, e dry has to be replaced by a new value to involve the effects of liquid on the energy dissipation in the solid.In the present work, we used the restitution coefficient of the impact on a 0.5 mm of water layer (e w ≈ 0.74) as the new reduction factor, because e w almost excludes the effect of liquid viscous force due to the small layer thickness and viscosity, and only represents the energy dissipation in the solid.
( 1) where St mc is the critical Stokes number calculated using Eq. ( 22).The predicted restitution coefficients are also plotted in Fig. 9. Apparently, it agrees much better with the experimental values than any other models.Only a bit of overestimations are observed at large St m .It should be closely related to the negligence of liquid inertia under high St m especially for the cases with large layer thicknesses.That is also one cause for the scatter of data in Fig. 9. Therefore, further work will aim at improving Eq. ( 24) to reduce the data scatter and enhance its predictability.

CONCLUSION
Aiming at normal impact between dry sphere and liquid layer with considerable thickness, the present paper extended elastohydrodynamic theory by coupling the physical force models to reveal the detailed impact mechanism.Based on the theoretical work, the following conclusions are drawn: (1) Three critical Stokes numbers, St mc1 , St dc and St mc2 are developed corresponding to the penetration, contact and rebound stages respectively.The Stokes numbers and critical values are compared to determine whether the sphere would rebound after the impact.(2) St mc1 decreases with increasing layer thickness while increases with sphere radius.Non-linear dependence of St mc2 on the layer thickness and sphere radius are observed.Increasing layer thickness would facilitate the sticking of sphere on the liquid layer.Moreover, the effects are enhanced by increasing the liquid viscosity.Opposite trends are observed regarding the effect of sphere radius.(3) When the layer thickness and liquid viscosity are changed within a low level, most of the kinetic energy is dissipated in the contact stage.However, with further increase of these two parameters, the energy dissipation in the penetration stage becomes predominant while the dissipation in the contact and rebound stage keeps decreasing till to zero.(4) An equation is proposed to estimate the restitution coefficient of the wet normal impact with considerable liquid layer thickness.It provides more accurate predictions than the existing models which are applicable for "thin-layer impact".
Fig.2plots a comparison between the drag coefficient estimated by Eqs.(9) and (10).Within the Reynolds numbers

Fig. 4 .
Fig. 4. Maximum length of liquid bridge, L max versus St m for normal impact with an impact velocity of 2.28 m s -1 .

Fig. 6 .
Fig. 6.Critical Stokes number for the penetration stage as a function of impact parameters.(a) liquid layer thickness; (b) sphere radius.

Fig. 9 .
Fig. 9. Comparison between experimental and theoretical restitution coefficients of normal impact as a function of St m .